A student wonders why her textbook disagrees with her solution to the domain of a composite function. Doctor Peterson walks through three approaches, as well as a method that checks her answer — and the book's.

The constant function f(x) = 0 will give zero no matter what function
g(x) it is integrated with. Does this mean that the constant function
zero is orthogonal to all functions? Also, what could be the geometrical
interpretation of orthogonal functions?

I am riding a bike and have several possible routes, each of which
contains various traffic lights, amounts of traffic, the possibility
of being stopped by a policeman for running a light, and other factors
which influence the time it will take to ride each route. How can I
model the routes to predict which will be the best choice?

I know that under specific conditions we can calculate the roots of a
function using the Newton-Raphson Method. The Newton-Raphson method
uses the tangent to approximate the root. I want to know if there is a
method to approximate the root with a parabola instead of the tangent.

Can you help me give a description of all Euclidean functions of Z?
The common example is of course the absolute value function, but it
seems to me that other weird Euclidean functions can be constructed, too.